1. กลุ่มที่ 1 เจตน์ & ดลยา
Probability
In the real world events can not be predicted with total certainty. The
How Likely
best we can do is say how likely they are to happen, using the idea of
probability.
When a coin is tossed, there are two possible outcomes:
Tossing a Coin
• heads (H) or
• tails (T)
We say that the probability of the coin landing H is 1/2.
Similarly, the probability of the coin landing T is 1/2.
When a single die is thrown, there are six possible
Throwing Dice
outcomes: 1, 2, 3, 4, 5, 6.
The probability of throwing any one of these numbers
is 1/6.
In general:
Probability
2. กลุ่มที่ 2 ตวงสิ ทธิ์ & เพญผกา
็
Probability Tree Diagrams
Calculating probabilities can be hard, sometimes you add them,
sometimes you multiply them, and often it is hard to figure out what to do
... tree diagrams to the rescue!
3. กลุ่มที่ 3 ยุทธนา & โซลี
There are basically two types of permutation:
Permutations
1. Repetition is Allowed: such as the lock above. It could be "333".
2. No Repetition: for example the first three people in a running race.
You can't be first and second.
5. กลุ่มที่ 5 สุ ทธินันท์ & ศิวรักษ์
Combinations
There are also two types of combinations (remember the order does not matter now):
1. Repetition is Allowed: such as coins in your pocket (5,5,5,10,10)
2. No Repetition: such as lottery numbers (2,14,15,27,30,33)
Actually, these are the hardest to explain, so I will come back to this later.
1. Combinations with Repetition
This is how lotteries work. The numbers are drawn one at a time, and if you have the
2. Combinations without Repetition
lucky numbers (no matter what order) you win!
The easiest way to explain it is to:
• assume that the order does matter (ie permutations),
• then alter it so the order does not matter.
Going back to our pool ball example, let us say that you just want to know which 3 pool
balls were chosen, not the order.
We already know that 3 out of 16 gave us 3,360 permutations.
But many of those will be the same to us now, because we don't care what order!
For example, let us say balls 1, 2 and 3 were chosen. These are the possibilites:
9. กลุ่มที่ 9 วนดี & ศิริรัตน์
ั
Complex Numbers
A complex number is made up of both real and imaginary
components. It can be represented by an expression of the form (a+bi),
where a and b are real numbers and i is imaginary. When defining i we say
that i = . Then we can think of i2 as -1. In general, if c is any positive
number, we would write:
.
If we have a complex number z, where z=a+bi then a would be the
real component (denoted: Re z) and b would represent the imaginary
component of z (denoted Im z). Thus the real component of z=4+3i is 4 and
the imaginary component would be 3. From this, it is obvious that two
complex numbers (a+bi) and (c+di) are equal if and only if a=c and b=d,
that is, the real and imaginary components are equal.
The complex number (a+bi) can also be represented by the ordered
pair (a,b) and plotted on a special plane called the complex plane or the
Argand Plane. On the Argand Plane the horizontal axis is called the real axis
and the vertical axis is called the imaginary axis.
10. กลุ่มที่ 10 ออมสิ น & ชฎาพร
Properties of the Complex Set
The set of complex numbers is denoted . Just like any other number set there are
rules of operation.
The sum and difference of complex numbers is defined by adding or subtracting their
real components ie:
The communitive and distributive properties hold for the product of complex
numbers ie:
When dividing two complex numbers you are basically rationalizing the
denominator of a rational expression. If we have a complex number defined as z =a+bi
then the conjuate would be . See the following example:
Example:
11. กลุ่มที่ 11 ชวลวทย์ & รัตนาภรณ์
ั ิ
Conjugates
The geometric inperpretation of a complex conjugate is the
reflection along the real axis. This can be seen in the figure below where z =
a+bi is a complex number. Listed below are also several properties of
conjugates.
Properties:
12. กลุ่มที่ 12 วศิน & ผกามาศ
Absolue Value/Modulus
The distance from the origin to any complex number is the absolute
value or modulus. Looking at the figure below we can see that Pythagoras'
Theorem gives us a formula to calculate the absolute value of a complex
number z = a+bi
And from this we get:
There are also some properties of absolute values dealing with complex
numbers. These are:
20. กลุ่มที่ 20 โกสิทธ์ิ
Difference of sets
The difference of sets “A-B” is the set of all elements of “A”, which do not belong to
“B”.
In the set builder form, the difference set is :
A − B = {x x ∈ A ∧ x ∉ B}
or A − B = A ∩ B'
The difference of sets “B - A” is the set of all elements of “B”, which do not belong
To “A”.
In the set builder form, the difference set is :
B − A = {x x ∈ B ∧ x ∉ A}
or B − A = B ∩ A'
